In this wiki, we will learn about differentiating logarithmic functions which are given by , in particular the natural logarithmic function using the differentiation rules . We can easily prove that these logarithmic functions are easily differentiable by looking at there graphs:
Proof: Let then by basic logarithmic properties we have
Differentiating the above equation implicitly with respect to and using we get
Prove the power rule using logarithmic differentiation.
First we state the power rule: for any real number ,
So first, we let then by logarithmic differentiation for we have
Using (as proved above), we have
which was to be proved.
We can write using logarithmic rules, and thus we have
Differentiating both the sides, we have
Using chain rule, we have
Using product rule, we have
Therefore, we have
First, let us try to differentiate the second term
Using product rule,
Thus, we have
which is what we needed to find.
This is where we need to directly use the quotient rule.
Using quotient rule, we have
We cannot directly approach this using differentiation rules. We need to bring suitable form for the function to be differentiated:
We now differentiate both sides with respect to using the chain rule on the left side and the product rule on the right: